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Country Boy

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  1. Given an open line segment, say (a, b), there are two ways to "compactify" it. One, the "Stone-Cech compactification", is to add the "endpoints", a and b, to get [a, b]. Another, the "one point compactification" is to imagine bending the line segment into a circle and adding a single point at the join. Following those ideas, we can "compactify" the set of all real numbers by adding "+ infinity" and "- infinity" or by adding a single point, "infinity", so that the topology becomes that of a circle, Similarly here are two different ways to "compactify" the open disk, [tex]\{(x, y)| x^2+ y^2< r^2\}[/tex]. The "Stone-Cech compactification", by adding the points on the boundary to get [tex]\{(x, y)| x^2+ y^2= r^2\}[/tex], and the "one point compactification" where you bend the disk up into a sphere, adding a single point at the top. We can do the same thing with the set of all complex numbers, adding an infinite number of "points at infinity", one in every direction, or add a single "point at infinity". In the first case the topology is that of a closed disk and in the second the topology is that of a sphere.
  2. The acceleration due to gravity is -g (which you are taking to be -10) vertically. Taking y to be positive upward and x positive to the right, the equations of motion are x= v_x t and y= v_y t- (g/2)t^2. The initial speed is 10 m/s at 30 degrees so v_x= 10(cos(30))= 10(1/2)= 5 m/s and v_y= 10(sin(30))= 10(sqrt(3)/2)= 5 sqrt(3) m/s which is about 8.7 m/s. So we have x= 5t and y= 8.7t- 5t^2. From x= 5t, t= x/5 so y= 8.7(x/5)- 5(x^2/25)= 1.74x- 0.2x^2. That is, of course, a parabola opening downward. The slope of the tangent line, at a given x, is 1.74- 0.4x. After 0.5 seconds, x= 5(0.5)= 2.5 and y= 8.7(0.5)- 5(0.25)= 5.6. The slope of the tangent line is 1.74- 0.4(2.5)= 0.74 so the tangent line to the trajectory at that time is y= 0.74(x- 2.5)+ 5.6. The tangential acceleration is the projection of the vector (0, -10) on that line.
  3. "It must not be vertebrate or fish". Fish are vertebrates!
  4. Help with what? What are you trying to do? Do you have a question?
  5. Country Boy

    Math

    Your question is simply too vague. Find the equation of a line given what information? Most common is "find the equation of the line through two given points". But you could also be asked to "find the equation of the line through a given point having a given slope" or "find the equation othf the line through a given point parallel to a given line" or "find the equation of the line through a given point perpendicular to a given line". Any (non-vertical) line can be written "y= ax+ b". If you are given two points you can put the x, y coordinates into that equation to get two equations to solve for a and b. For example, if the two points are (3, 5) and (7, 9) then we have 5= 3a+ b and 9= 7a+ b. Subtracting the first equation from the second, 4= 4a so a= 1. Then 5= 3+ b so b= 5- 3= 2. The line is given by y= x+ 2. If you are given the point (3, 5) and slope 3 then, because the "a" in "y= ax+ b" is the slope, a= 3. Putting the x= 3, y= 5 in y= 3x+ b we have 5= 3(3)+ b so b= 5- 9= -4. The line is given by y= 3x- 4. If you are given the point (3, 5) and are told that the line is parallel to the line y= 7x+ 9 then the slope is the same as the slope of the given line, 7, so we have the previous problem: 5= 7(3)+ b. b= 5- 21= -16. The line is given by y= 7x- 16. If you are given the point (3, 6) and are told that the line is perpendicular to the line y= 4x- 5 then the slope is the negative reciprocal of the slope of the given line, -1/4, so we gave 6= (-1/4)(3)+ b. b= 6+ 3/4= 27/4. The line is given by y= (-1/4)x+ 27/4. We can also write that as 4y= 27- x or x+ 4y= 27. (Any vertical line can be written x= constant. The x-value of a given point on the line gives you the constant.)
  6. You need to learn definitions and basic concepts before you can understand formulas! In your first post you ask for "slope of points(2,2) and (3,5)". Points don't have a slope! That was one reason the first response asked you to state whole problem. Most likely you were asked to "find the slope of the line through points (2, 2) and (3,5)". Whoever gave you this problem clearly expects you to know what the word "slope" means here! What definition of "slope of a line" were you given?
  7. I am tempted to say "'second' comes between 'first' and 'third'".
  8. Draw a line from one of the vertices to the midpoint of the opposite side. What can you say about the two triangles created?
  9. According to Wikipedia, "Epigenetics most often denotes changes that affect gene activity and expression, but can also be used to describe any heritable phenotypic change" so, yes epigenetic changes affect genes and can be passed on to off spring. However, "working out" and "eating lots of nutrient dense food" are NOT "epigenetic". They are phenotypic, do not affect genes, and are NOT passed on to offspring.
  10. It doesn't have to be prime- as long as a number is not a perfect square or divisible by a perfect square, you cannot continue. For example to simplify the square root of 216, I can observe 216= 2*2*2*3*3*3= 2^3 3^3 (that's its "prime factorization"). Since I want the square root, I look for squares- powers of 2: (2*2)(2)(3*3)(3)= 4(2)(9)(3). "4" and "9" are "perfect squares", 2 squared and 3 squared. [math]\sqrt{216}= \sqrt{4(9)(2)(3)= \sqrt{4}\sqt{9}\sqrt{2(3)}= 2(3)\sqrt{6}= 6\sqrt{6}[/math]. "6" is not prime but it is not a perfect square either. Notice that 2*2*2= 8 and 3*3*3= 27 are "perfect cubes", $2^3$ and $3^3$ so its cube root: [math]\sqrt[3]{216}= \sqrt[3]{2^3(3^3}= 2(3)= 6[/math].
  11. Originally Newton tried to define the derivative as "dy divided by dx" where dy and dx are "infinitesmals" but was not able to give a rigorous definition of "infinitesimal". The Bishop Berkeley famously satirized them as "ghosts of vanished quantities". Later people like Cauchy redefined the derivative in terms of limits. But recently Abraham Robinson and others resurrected "infinitesmals" by extending the real numbers to include both "infinite" and "infinitesmals" in "non-standard analysis": https://en.wikipedia.org/wiki/Non-standard_analysis.
  12. When an object is sitting on the ground there are two forces- the force of gravity pointing downward and the force due to the pressure the ground is exerting on the object. Those are "equal and opposite", they cancel and one could, as well, say there is no (net) force. If an object is sitting on a frictionless tilted surface such as a wedge there is the force of gravity but now the force due to the surface is not directly opposite to gravitational force and we may have motion along the surface. In "real life", with friction, the friction force is parallel to the surface so we can think of this as three forces- the vertical force of gravity, the pressure force due to the surface which is perpendicular to the surface, and friction which is parallel to the surface.
  13. Do you not know how 'google' works? I entered "parts working together for one purpose" in Google and immediately got "system".
  14. If the weight of herbs alone is "m" and the weight of herbs and vodka is "M" then the weight of vodka alone is M-m so that the ratio of "herbs to vodka" is m/(M- m).
  15. No, I don't know what you by "peak state". If you know what you mean please tell us!
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